Skip to main content
Log in

Effect of thoracic muscle on dynamic performance of flexible flapping wings of insects

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

The thorax is the central component of the insect flight drivetrain, making it essential to understand how thoracic muscles influence insect flight for flapping wing microair vehicle design. This paper presents a theoretical model that takes into account the influence of thoracic muscles on flapping wing motion with reference to real insects. The thoracic muscle effect is simulated by the chordwise torsional spring, whose stiffness is derived from a comparison test with the results of real insect experiments. The elastic deformation of the flexible flapping wing is modeled by the von Kármán nonlinear plate theory. The predictive quasi-steady aerodynamic model based on the blade element theory can estimate the aerodynamic force, and the modeling of the spanwise bending and twisting is done with quadratic polynomials. The equations of motion are solved using the Newmark-Raphson method. Results suggest that including the influence of thoracic muscles decreases cycle-averaged lift and power, but enhances the efficiency of lift production by 23%. Moreover, it also postpones pitching motion and reduces its amplitude, though the movement trends of the flapping motion remain approximately unchanged regardless of the inclusion of the thoracic muscle effect.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Chirarattananon, P., Ma, K. Y., Wood, R. J.: Adaptive control of a millimeter-scale flapping-wing robot. Bioinspir. Biomim. 9(2), 025004 (2014).

  2. Phan, H. V., Kang, T., Park, H. C.: Design and stable flight of a 21 g insect-like tailless flapping wing micro air vehicle with angular rates feedback control. Bioinspir. Biomim. 12(3), 036006 (2017).

  3. Karásek, M., Muijres, F.T., De Wagter, C., Remes, B.D., De Croon, G.C.: A tailless aerial robotic flapper reveals that flies use torque coupling in rapid banked turns. Science 361(6407), 1089–1094 (2018)

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  4. Jafferis, N.T., Helbling, E.F., Karpelson, M., Wood, R.J.: Untethered flight of an insect-sized flapping-wing microscale aerial vehicle. Nature 570(7762), 491–495 (2019)

    Article  ADS  CAS  PubMed  Google Scholar 

  5. Phan, H.V., Aurecianus, S., Kang, T., Park, H.C.: KUBeetle-S: An insect-like, tailless, hover-capable robot that can fly with a low-torque control mechanism. Int. J. Micro Air Vehicles. 11, 1756829319861371 (2019)

    Article  Google Scholar 

  6. Ellington, C.: The aerodynamics of hovering insect flight. III. Kinematics. Philos. Trans. R. Soc. Lond. B Biol. Sci. 305(1122), 41–78 (1984)

  7. Sane, S.P., Dickinson, M.H.: The control of flight force by a flapping wing: lift and drag production. J. Exp. Biol. 204(Pt 15), 2607–2626 (2001)

    Article  CAS  PubMed  Google Scholar 

  8. Shyy, W., Aono, H., Kang, C.K., Liu, H., An introduction to flapping wing aerodynamics, Cambridge University Press (2013)

  9. Dickinson, M.H., Lehmann, F.O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science 284(5422), 1954–1960 (1999)

    Article  CAS  PubMed  Google Scholar 

  10. Liu, H.: Computational biological fluid dynamics: digitizing and visualizing animal swimming and flying. Integr. Comp. Biol. 42(5), 1050–1059 (2002)

    Article  PubMed  Google Scholar 

  11. Ramamurti, R., Sandberg, W.C.: A three-dimensional computational study of the aerodynamic mechanisms of insect flight. J. Exp. Biol. 205(Pt 10), 1507–1518 (2002)

    Article  PubMed  Google Scholar 

  12. Sun, M., Tang, J.: Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion. J. Exp. Biol. 205(Pt 1), 55–70 (2002)

    Article  PubMed  Google Scholar 

  13. Usherwood, J. R., Ellington, C. P.: The aerodynamics of revolving wings II. Propeller force coefficients from mayfly to quail. J. Exp. Biol. 205(Pt 11), 1565–1576 (2002)

  14. Prempraneerach, P., Hover, F., Triantafyllou, M. S., The effect of chordwise flexibility on the thrust and efficiency of a flapping foil, in: Proc. 13th Int. Symp. on Unmanned Untethered Submersible Technology: special session on bioengineering research related to autonomous underwater vehicles, New Hampshire, 2003, pp. 152–170

  15. Wang, Z.J.: The role of drag in insect hovering. J. Exp. Biol. 207(Pt 23), 4147–4155 (2004)

    Article  PubMed  Google Scholar 

  16. Wang, Z.J., Birch, J.M., Dickinson, M.H.: Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments. J. Exp. Biol. 207(Pt 3), 449–460 (2004)

    Article  PubMed  Google Scholar 

  17. Ramamurti, R., Sandberg, W.C.: A computational investigation of the three-dimensional unsteady aerodynamics of Drosophila hovering and maneuvering. J. Exp. Biol. 210(Pt 5), 881–896 (2007)

    Article  PubMed  Google Scholar 

  18. Bhat, S. S., Zhao, J. S., Sheridan, J., Hourigan, K., Thompson, M. C.: Effects of flapping-motion profiles on insect-wing aerodynamics. J. Fluid Mech. 884 (2020)

  19. Zhao, L., Huang, Q., Deng, X., Sane, S.P.: Aerodynamic effects of flexibility in flapping wings. J. R. Soc. Interface. 7(44), 485–497 (2010)

    Article  PubMed  Google Scholar 

  20. Nakata, T., Liu, H.: Aerodynamic performance of a hovering hawkmoth with flexible wings: a computational approach. Proc. Biol. Sci. 279(1729), 722–731 (2012)

    PubMed  Google Scholar 

  21. Le, T. Q., Truong, T. V., Park, S. H., Quang Truong, T., Ko, J. H., Park, H. C., Byun, D.: Improvement of the aerodynamic performance by wing flexibility and elytra–hind wing interaction of a beetle during forward flight. Journal of the Royal Society Interface. 10(85), 20130312 (2013)

  22. Chen, L., Wang, L., Wang, Y.Q.: Efficient Fluid-Structure Interaction Model for Twistable Flapping Rotary Wings. AIAA J. 60(12), 6665–6679 (2022)

    Article  ADS  Google Scholar 

  23. Nakata, T., Liu, H., Tanaka, Y., Nishihashi, N., Wang, X., Sato, A.: Aerodynamics of a bio-inspired flexible flapping-wing micro air vehicle. Bioinspir Biomim. 6(4), 045002 (2011)

  24. Kim, D. K., Lee, J. S., Lee, J. Y., Han, J. H., An aeroelastic analysis of a flexible flapping wing using modified strip theory, in: Active and Passive Smart Structures and Integrated Systems 2008, SPIE, 2008, pp. 477–484

  25. Yin, B., Luo, H. X.: Effect of wing inertia on hovering performance of flexible flapping wings. Physics of Fluids. 22(11), 111902 (2010)

  26. Addo-Akoto, R., Han, J.-S., Han, J.-H.: Aerodynamic characteristics of flexible flapping wings depending on aspect ratio and slack angle. Phys Fluids. 34(5), 051911 (2022)

  27. Chen, L., Yang, F.L., Wang, Y.Q.: Analysis of nonlinear aerodynamic performance and passive deformation of a flexible flapping wing in hover flight. J. Fluids Struct. 108, 103458 (2022)

    Article  ADS  Google Scholar 

  28. Wang, C., Tang, H., Zhang, X.: Fluid-structure interaction of bio-inspired flexible slender structures: a review of selected topics. Bioinspir Biomim. 17(4) (2022)

  29. Reade, J., Jankauski, M.: Investigation of chordwise functionally graded flexural rigidity in flapping wings using a two-dimensional pitch–plunge model. Bioinspir Biomim. 17(6), 066007 (2022)

  30. Zhao, M., Zou, Y., Fu, Q., He, W.: Effects of airfoil on aerodynamic performance of flapping wing. Biomimetic Intell. Robot. 1, 100004 (2021)

    Article  Google Scholar 

  31. Grimaldi, D., Engel, M. S., Engel, M. S., Engel, M. S., Evolution of the Insects, Cambridge University Press (2005)

  32. Ennos, A.R.: A comparative study of the flight mechanism of Diptera. J. Exp. Biol. 127(1), 355–372 (1987)

    Article  Google Scholar 

  33. Chattaraj, N., Ganguli, R.: Mechatronic approaches to synthesize biomimetic flapping-wing mechanisms: a review. Int. J. Aeronaut. Space Sci. 24(1), 105–120 (2023)

    Article  Google Scholar 

  34. Jankauski, M. A.: Measuring the frequency response of the honeybee thorax. Bioinspir Biomim. 15(4), 046002 (2020)

  35. Casey, C., Heveran, C., Jankauski, M.: Experimental studies suggest differences in the distribution of thorax elasticity between insects with synchronous and asynchronous musculature. Journal of the Royal Society Interface. 20(201), 20230029 (2023)

  36. Lynch, J., Gau, J., Sponberg, S., Gravish, N.: Dimensional analysis of spring-wing systems reveals performance metrics for resonant flapping-wing flight. J. R. Soc. Interface. 18(175), 20200888 (2021)

  37. Pons, A., Beatus, T.: Distinct forms of resonant optimality within insect indirect flight motors. Journal of the Royal Society Interface. 19(190), 20220080 (2022)

  38. Karásek, M.: Robotic Hummingbird: Design of a Control Mechanism for a Hovering Flapping Wing Micro Air Vehicle. Universite libre de Bruxelles, Bruxelles, Belgium (2014)

    Google Scholar 

  39. Lindsay, T., Sustar, A., Dickinson, M.: The function and organization of the motor system controlling flight maneuvers in flies. Curr. Biol. 27(3), 345–358 (2017)

    Article  CAS  PubMed  Google Scholar 

  40. Grimaldi, D., Engel, M. S., Evolution of the Insects, Cambridge University Press (2005)

  41. Snodgrass, R. E., Principles of insect morphology, Cornell University Press (2018)

  42. Chin, D.D., Lentink, D.: Flapping wing aerodynamics: from insects to vertebrates. J. Exp. Biol. 219(Pt 7), 920–932 (2016)

    Article  PubMed  Google Scholar 

  43. Sun, M.: Insect flight dynamics: stability and control. Rev. Mod. Phys. 86(2), 615–646 (2014)

    Article  ADS  Google Scholar 

  44. Yang, F.L., Chen, L., Wang, Y.Q.: Improved model for flexible flapping wings: considering spanwise twisting and bending. AIAA J. 60(12), 6680–6691 (2022)

    Article  ADS  Google Scholar 

  45. Trahair, N.S.: Nonlinear elastic nonuniform torsion. J. Struct. Eng. Asce. 131(7), 1135–1142 (2005)

    Article  Google Scholar 

  46. Wang, Q., Goosen, J. F. L., van Keulen, F.: A predictive quasi-steady model of aerodynamic loads on flapping wings. J. Fluid Mech. 800 688–719 (2016)

  47. Phan, H. V., Park, H. C.: Wing inertia as a cause of aerodynamically uneconomical flight with high angles of attack in hovering insects. J. Exp. Biol. 221(19), jeb187369 (2018)

  48. Combes, S. A., Daniel, T. L.: Flexural stiffness in insect wings. I. Scaling and the influence of wing venation. J. Exp. Biol. 206(Pt 17), 2979–2987 (2003)

  49. Wang, Q., Goosen, J., van Keulen, F.: An efficient fluid–structure interaction model for optimizing twistable flapping wings. J. Fluid Struct. 73, 82–99 (2017)

  50. Berman, G. J., Wang, Z. J.: Energy-minimizing kinematics in hovering insect flight. J. Fluid Mech. 582 153–168 (2007)

  51. Willmott, A. P., Ellington, C. P.: The mechanics of flight in the hawkmoth Manduca sexta. II. Aerodynamic consequences of kinematic and morphological variation. J. Exp. Biol. 200(Pt 21), 2723–2745 (1997)

  52. Willmott, A. P., Ellington, C. P.: The mechanics of flight in the hawkmoth Manduca sexta I. Kinematics of hovering and forward flight. J. Exp. Biol. 200(21), 2705–2722 (1997)

  53. Nakata, T., Liu, H.: A fluid–structure interaction model of insect flight with flexible wings. J. Comput. Phys. 231(4), 1822–1847 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  54. Wu, J.H., Sun, M.: Unsteady aerodynamic forces of a flapping wing. J. Exp. Biol. 207(7), 1137–1150 (2004)

    Article  PubMed  Google Scholar 

Download references

Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant Nos. 12272088 and 11922205).

Author information

Authors and Affiliations

Authors

Contributions

FLY helped in methodology, data curation, formal analysis, writing—original draft, visualization, investigation, software, validation. YQW contributed to supervision, conceptualization, funding acquisition, project administration, writing—review and editing, resources.

Corresponding author

Correspondence to Yan Qing Wang.

Ethics declarations

Conflict of interest

All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or nonfinancial interest in the subject matter or materials discussed in this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, F.L., Wang, Y.Q. Effect of thoracic muscle on dynamic performance of flexible flapping wings of insects. Acta Mech 235, 597–613 (2024). https://doi.org/10.1007/s00707-023-03757-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-023-03757-2

Navigation